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Invariant random subgroups of linear groups

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Abstract

Let Γ < GL n (F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty topology. An invariant random subgroup (IRS) of Γ is a conjugation invariant Borel probability measure on Sub(Γ). An IRS is called non-trivial if it does not have an atom in the trivial group, i.e. if it is non-trivial almost surely. We denote by IRS0(Γ) the collection of all non-trivial IRS on Γ.

Theorem 0.1: With the above notation, there exists a free subgroup F < Γ and a non-discrete group topology on Γ such that for every μ ∈ IRS0(Γ) the following properties hold:

μ-almost every subgroup of Γ is open

  • F ·Δ = Γ for μ-almost every Δ ∈ Sub(Γ).

  • F ∩ Δ is infinitely generated, for every open subgroup. In particular, this holds for μ-almost every Δ ∈ Sub(Γ).

  • The map

Φ: (Sub(Γ), μ) → (Sub(F),Φ*μ) Δ → Δ ∩ F is an F-invariant isomorphism of probability spaces.

A more technical version of this theorem is valid for general countable linear groups. We say that an action of Γ on a probability space, by measure preserving transformations, is almost surely non-free (ASNF) if almost all point stabilizers are non-trivial.

Corollary 0.2: Let Γ be as in the Theorem above. Then the product of finitely many ASNF Γ-spaces, with the diagonal Γ action, is ASNF.

Corollary 0.3: Let Γ < GLn(F) be a countable linear group, A Δ Γ the maximal normal amenable subgroup of Γ — its amenable radical. If μ ∈ IRS(Γ) is supported on amenable subgroups of Γ, then in fact it is supported on Sub(A). In particular, if A(Γ) = <e> then Δ = <e>, μ almost surely.

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er Sheva, 84105, Israel

    Yair Glasner

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With an appendix by Tsachik Gelander1 and Yair Glasner

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Glasner, Y. Invariant random subgroups of linear groups. Isr. J. Math. 219, 215–270 (2017). https://doi.org/10.1007/s11856-017-1479-x

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